A. Industrial Setting
Numerous industries employ processes which require accurate delivery of a binary gas mixture consisting of a gas of interest and a carrier gas. To achieve accurate delivery, these industries require precise measurements of the concentration of the gas of interest in the flowing gas mixture, where the gas of interest is typically of high purity and may be highly corrosive. Examples of these processes include: chemical vapor deposition, dopant diffusion in, for example, the semiconductor industry, etching, the operation of high efficiency hydrogen cooled generators, and the like.
Current practice in the above and other industries is to use mass flow controllers upstream of a bubbler (vaporizer) to predict (control) the concentration of the binary gas mixture which is generated in the bubbler. This approach suffers from insufficient accuracy due to, among other things: variations in the bubbler temperature; instability of the temperature and pressure of the binary gas mixture; possible leakages in the gas lines upstream and downstream of the bubbler; and concentration time delays between the mass flow controllers and the points of interest, especially at low flow rates.
In addition, the existing equations used to predict a bubbler's pick-up rate are inaccurate. The following is an example of such an equation, where G.sub.A is the mass pickup rate of gas A (the gas of interest), Q.sub.B is a flow rate of gas B (the carrier gas), P is the pressure of the binary gas mixture, P.sub.VA is the vapor pressure of gas A at the bubbler's operating temperature, M is the molecular weight of gas A, T is the temperature of the binary gas mixture, and R is the ideal gas constant: ##EQU1## This equation can exhibit an inaccuracy as high as 20% when the bubbler's operating temperature is well below the boiling point of gas A, which is the typical operating condition used in practice.
At present there is a wide variety of concentration sensors on the market which use different measurement approaches, including acoustical (EPISON and MINISON devices sold by Thomas Swan of the United Kingdom), optical (IR-5 device sold by MKS Instruments, Inc., Andover, Mass.), thermal conductivity (Varian Model 3400 Gas Chromatograph, sold by Varian Vacuum Products, Lexington, Mass.), and mass spectroscopy. None of these approaches fulfill all the requirements for a binary gas measuring system, including robustness, maintenance free operation, and the ability to produce highly accurate and repeatable real time concentration measurements of high purity and/or highly corrosive gaseous media.
B. Concentration Sensors Which Employ Acoustical Energy
As discussed above, acoustical measurements have been used in the past in concentration sensors. At their heart, such devices involve a measurement of the speed of sound (or, more accurately, the speed of propagation of acoustical energy) in a medium, with variations in the measured speed being indicative of variations in the concentration of the chemical of interest.
There are two main approaches for measuring the speed of sound in a medium, namely, the phase approach and the pulse approach. The phase approach gives a precise measurement of the phase velocity of acoustical energy in the medium, either by means of a fixed-frequency, variable-path, cylindrical acoustic interferometer or by means of a variable-frequency, fixed-path, spherical acoustic resonator. In accordance with this approach, the speed of sound is assumed to be the same as the phase velocity, which, as discussed below, is not always the case.
The pulse approach provides a direct measurement of wavefront velocity (i.e., speed of sound) and can be implemented either in a shadow format with a separate transmitter and receiver, or an echo format with only one transducer which fulfills both the transmitting and receiving functions (see U.S. Pat. No. 5,325,703). The pulse approach as practiced in the prior art has generally been less accurate than the phase approach.
(1) Phase Approach
Phase methods employ a continuous acoustic wave. The principles of the propagation of sound in finite cylindrical and spherical resonators were worked out by Lord Rayleigh and set forth in his 1877 treatise "The Theory of Sound" (Dover Publ., New York, 1945). From a practical point of view, neither a spherical acoustic resonator nor a cylindrical acoustic interferometer can be used widely for several practical reasons.
For real gases with speeds of sound in the range of 100-1500 meters/second, a spherical resonator requires an efficient broad band acoustic transducer, which does not exist at present. State of the art principles of design of such transducers for liquids are known (see Desilets et al. "The Design of Efficient Broad--Band Piezoelectric Transducers", IEEE Transactions on Sonics and Ultrasonics, Vol. SU-25, No. 3, May 1978, 115-125 and Takeshi Inoue et al. "Design of Ultrasonic Transducers with Multiple Acoustic Matching Layers for Medical Application", IEEE Transaction on Ultrasonics, Ferroelectrics and Frequency Control, Vol. UFFC-34, No. 1, January 1987, 8-16), but their capabilities are very weak for concentration measurements in gaseous media.
A cylindrical acoustic interferometer requires a precision mechanical system with a known distance between a transducer and a moving piston (see Potzick "On the Accuracy of Low Flow Rate Calibrations at the National Bureau of Standards", ISA Transactions, Vol. 25, No. 2, 1986, 19-23). Such a device thus requires continuous maintenance.
In addition to these specific drawbacks, it is important to note that both phase methods use phase velocity as a measure of the speed of sound. Phase velocity and the speed of sound can be different at different frequencies for different gases due to relaxation processes (see Gooberman "Ultrasonics", Hart Publishing Company, Inc., NYC, 1969) and/or shock waves (see Elmore et al. "Physics of Waves", Dover Publications, Inc., NY, 1969). These effects are capable of producing incorrect values for the speed of sound, and, in turn, incorrect concentration measurements.
(2) Pulse Approach
Pulse acoustical concentration measurement methods and instruments for testing binary gases are known. Publications in this area include:
(a) E. Polturak, S. Garrett and S. Lipson, "Precision acoustic gas analyzer for binary mixtures", Rev. Sci. Instrum. 57 (11), American Institute of Physics, November 1986, 2837-2841. PA1 (b) G. Cadet, J. Valdes and J. Mitchell, "Ultrasonic time-of-flight method for on-line quantitation of semiconductor gases", Ultrasonics Symposium, Proceedings, New York, Institute of Electrical and Electronic Engineers, 1991, 295-300. PA1 (c) G. Hallewell and L. Lynnworth, "A simplified formula for the analysis of binary gas containing a low concentration of a heavy vapor in a lighter carrier", Ultrasonics Symposium, Proceedings, New York, Institute of Electrical and Electronic Engineers, 1994, 1311-1316. PA1 (d) J. P. Stagg, "Reagent Concentration Measurements in Metal Organic Vapour Phase Epitaxy (MOVPE) Using an Ultrasonic Cell", Chemtronics, Vol. 3, March 1988, Harlow, Essex, UK, 44-49. PA1 (e) G. Hallewell, G. Crawford, D. McShurley, G. Oxoby and R. Reif, "A Sonar-Based Technique for the Ratiometric Determination of Binary Gas Mixtures", Nuclear Instruments and Methods in Physics Research, A264, 1988, North-Holland, Amsterdam, 219-234. PA1 (f) M. Joos, H. Muller and G. Lindner, "An ultrasonic sensor for the analysis of binary gas mixtures", Sensors and Actuators, B. 15-16, 1993, 413-419. PA1 (g) L. Zipser and F. Wachter, "Acoustic sensor for ternary gas analysis", Sensors and Actuators, B. 26-27, 1995, 195-198. PA1 (h) U.K. Patent GB 2,215,049. PA1 (i) U.S. Pat. Nos. 4,596,133; 4,630,482; 4,850,220; 5,060,507; 5,325,703; 5,351,522; 5,369,979; and 5,392,635. PA1 (a) a chamber for receiving the gas; PA1 (b) a transmitter which, during use of the apparatus, transmits a pulse of acoustical energy through the gas in the chamber, said transmitter having a characteristic dimension and said pulse of acoustical energy comprising a plurality of oscillations of the acoustical energy, said plurality of oscillations having a center frequency f.sub.c ; and PA1 (c) a receiver which, during use of the apparatus, detects the pulse after the pulse has passed through the gas in the chamber, said passing of the pulse through the gas defining an acoustical path length L.sub.p ; PA1 (a) generating a pulse of acoustical energy, said generation defining a near-field zone; PA1 (b) passing the pulse through the gas; and PA1 (c) detecting the pulse after its passage through the gas; PA1 0.5 megahertz.ltoreq.f.sub.c .ltoreq.10 megahertz; and/or PA1 L.sub.p .ltoreq.0.5 L.sub.n. PA1 (a) a transmitter which, during use of the apparatus, produces a pulse of acoustical energy which moves in alternating directions between first and second spaced apart points in the gas, liquid, or solid; PA1 (b) a receiver which, during use of the apparatus, detects the pulse; and PA1 (c) detecting means associated with the receiver for detecting the pulse after the pulse has made n passages between the first and second spaced apart points (the n-passage pulse) and after it has made m passages between those points (the m-passage pulse), where a passage constitutes movement of the pulse between the first and second spaced apart points in either direction, m and n are integers, and EQU m-n=2i PA1 where i is an integer. PA1 (a) producing a pulse of acoustical energy which moves in alternating directions between first and second spaced apart points in the gas, solid, or liquid; and PA1 (b) detecting the pulse after it has made n passages between the first and second spaced apart points (the n-passage pulse) and after it has made m passages between those points (the m-passage pulse), where a passage constitutes movement of the pulse between the first and second spaced apart points in either direction, m and n are integers, and EQU m-n=2i PA1 where i is an integer. PA1 (i) equalizing means for substantially equalizing the amplitudes of the n-passage and m-passage pulses; and PA1 (ii) identifying means for identifying reference points for the amplitude equalized n-passage and m-passage pulses. PA1 (i) the equalizing means produces an n-passage pulse whose maximum amplitude is U.sub.n and a m-passage pulse whose maximum amplitude is U.sub.m, where U.sub.n. substantially equals U.sub.m ; PA1 (ii) the identifying means comprises means for producing rectified envelopes of the amplitude equalized pulses; and PA1 (iii) the reference points for the amplitude equalized n-passage and m-passage pulses comprise those points where the rectified envelopes of those pulses have amplitudes of kU.sub.n and kU.sub.m, respectively, where PA1 0&lt;k&lt;1. PA1 (a) cross-correlating means associated with the receiver for performing a cross-correlation between a waveform representing the original pulse and the detected pulse, said cross-correlating means producing a cross-correlation function for the detected pulse; and PA1 (b) peak determining means for determining the peak of said cross-correlation function, said peak being indicative of the time the detected pulse was detected by the receiver. PA1 (a) producing a pulse of acoustical energy which passes through the gas, solid, or liquid (the "original pulse"); PA1 (b) detecting the pulse after it has passed through the gas, solid, or liquid (the "detected pulse"); PA1 (c) performing a cross-correlation between a waveform representing the original pulse and the detected pulse, said cross-correlation producing a cross-correlation function for the detected pulse; and PA1 (d) determining the peak of said cross-correlation function, said peak being indicative of the time the detected pulse was detected in step (b). PA1 (i) performing a fast Fourier transform of the detected pulse to produce a Fourier transformed detected pulse; PA1 (ii) multiplying the Fourier transformed detected pulse by the complex conjugate of a Fourier transform of the waveform representing the original pulse (the "template waveform") to produce a Fourier space cross-correlation function; and PA1 (iii) performing an inverse fast Fourier transform of said Fourier space cross-correlation function determined in step (ii) and taking the real part of said inverse fast Fourier transform to produce said cross-correlation function. PA1 (i) identifying the largest amplitude of the cross-correlation function and two amplitudes adjacent to the largest amplitude; PA1 (ii) fitting a second order polynomial to said three amplitudes; and PA1 (iii) determining the peak of the cross-correlation function from said second order polynomial. PA1 (i) windowed whereby the leading and trailing cycles of the pulse taper from zero to full amplitude; and/or PA1 (ii) frequency modulated. PA1 (a) a transmitter which, during use of the apparatus, produces acoustical energy which passes through the gas; PA1 (b) a receiver which, during use of the apparatus, detects the acoustical energy; and PA1 (c) computing means for determining the property of the gas based on the equations: EQU n.sub.f =6(n.sub.a -1); PA1 and EQU C=((2+n.sub.f)(kT)/(n.sub.f m.sub.m)).sup.1/2 PA1 where C is the speed of sound in the gas, T is the temperature of the gas, and k is Boltzmann's constant. PA1 (a) producing acoustical energy which passes through the gas; PA1 (b) detecting the acoustical energy after it has passed through the gas; and PA1 (c) determining the property of the gas based on the above equations. PA1 (a) a transmitter which, during use of the apparatus, produces acoustical energy which passes through the gas; PA1 (b) a receiver which, during use of the apparatus, detects the acoustical energy; and PA1 (c) computing means for determining the property of the gas based on the equation: ##EQU15## where C.sub.M is the speed of sound in the mixture, T is the temperature of the mixture, R is the universal gas constant, and x.sub.i, c.sub.Pi, M.sub.i, and z.sub.i are the volumetric concentration, specific heat, molecular weight, and compressibility factor, respectively, of the i.sup.th molecular species. PA1 (a) producing acoustical energy which passes through the gas; PA1 (b) detecting the acoustical energy after it has passed through the gas; and PA1 (c) determining the property of the gas based on the above equation. PA1 (a) introducing a gas, solid, or liquid into the test chamber, said gas, solid, or liquid having a known speed of sound C.sub.N ; PA1 (b) producing a pulse of acoustical energy which moves in alternating directions between first and second spaced apart points in the gas, solid, or liquid; PA1 (c) detecting the pulse after it has made n passages between the first and second spaced apart points (the n-passage pulse) and after it has made m passages between those points (the m-passage pulse), where a passage constitutes movement of the pulse between the first and second spaced apart points in either direction, m and n are integers, and EQU m-n=2i PA1 where i is an integer; and PA1 (c) determining the distance L between the first and second spaced apart points from the equation: ##EQU16## where t.sub.2 and t.sub.1 are the detection times of the m-passage and n-passage pulses, respectively. PA1 (a) evacuating the test chamber; PA1 (b) activating the transmitter to produce a pulse of acoustical energy and recording the resulting signal at the receiver; PA1 (c) repeating step (b) to produce a plurality of recorded signals; and PA1 (d) averaging the plurality of recorded signals to produce a stray path signal. PA1 (a) carrier gas flow control means (e.g., a mass flow controller) for controlling the flow of the carrier gas; PA1 (b) mixing means (e.g., a bubbler) for producing the mixture of the gas of interest and the carrier gas; PA1 (c) determining means for determining a property of the mixture, said property being indicative of the composition of the mixture (e.g., the property can be the mass flow of the gas of interest provided by the mixture); and PA1 (d) control means operatively connected to the carrier gas flow control means and the determining means for controlling the carrier gas flow control means based on a difference between the property of the mixture determined by the determining means and a desired value for that property. PA1 (a) determining a property of the mixture (e.g., the mass flow of the gas of interest provided by the mixture); and PA1 (b) controlling the flow of the carrier gas by determining the difference between the property of the mixture determined in step (a) and a desired value for that property.
C. Deficits of Existing Concentration Sensors Which Employ Acoustical Energy
None of the prior art acoustical methods perform satisfactory at present. In particular, none of these methods are able to fully meet the fundamental requirements of providing highly accurate concentration measurements without the need for frequent maintenance.
There are a variety of sources of significant error in acoustical concentration measurements. The most important of these are:
(1) Ambiguities in the existing mixing rules used to calculate concentrations from measured values of the speed of sound.
In all of the prior art publications, concentration values are calculated based on the known ideal gas equation for the speed of sound (see, for example, Polturak et al. "Precision acoustic gas analyzer for binary mixtures", Rev. Sci. Instrum. 57 (11), American Institute of Physics, November 1986, 2837-2841 and Cadet et al. "Ultrasonic time-of-flight method for on-line quantitation of semiconductor gases", Ultrasonics Symposium, Proceedings, New York, Institute of Electrical and Electronic Engineers, 1991, 295-300). That is, the calculation of concentration values is ultimately based on the following equation for the speed of sound C in a gas: ##EQU2## where ##EQU3## and where c.sub.p and c.sub.v are the specific heat capacities of the gas at constant pressure and constant volume, respectively, R=8.3144 is the universal gas constant, T is the gas temperature in .degree.K, and M is the gas' molecular weight. As discussed in Reid et al. "The Properties of Gases and Liquids", Fourth Edition, McGraw-Hill, Inc., New York, 1987, c.sub.p can be obtained from the formula: ##EQU4## where the coefficients A.sub.c, B.sub.c, C.sub.c, and D.sub.c are constants whose values depend on the chosen gas.
For a mixture M of two gases A and B, equation (1) can be written: ##EQU5##
The challenge in using equation (4) is, of course, determining appropriate values for .gamma..sub.M (the "mixture specific heat ratio") and M.sub.M (the "mixture molecular weight"). Different workers in the art have suggested a variety of mixing rules, which can lead to significantly different values for M.sub.M and .gamma..sub.M.
For example, to estimate M.sub.M, some publications have proposed simple averaging (see Joos et al. "An ultrasonic sensor for the analysis of binary gas mixtures", Sensors and Actuators, B. 15-16, 1993, 413-419 and Zipser et al. "Acoustic sensor for ternary gas analysis", Sensors and Actuators, B. 26-27, 1995, 195-198): ##EQU6## where x.sub.i is the volumetric concentration of gas i in the total volume.
On the other hand, more sophisticated mixing rules have been proposed, such as that of Chung et al., "Applications of kinetic gas theories and multiparameter correlation for prediction of dilute gas viscosity and thermal conductivity," Industrial & Engineering Chemistry Fundamentals, 1984, 23:8-13; Reid et al. "The Properties of Gases and Liquids", Fourth Edition, McGraw-Hill, Inc., New York, 1987, 413-414): ##EQU7## where T.sub.ci and V.sub.ci are the critical temperature and critical volume of gas i, respectively.
By substitution of the parameters for a 70:30 hydrogen sulfide/ethyl ether mixture into equations (5) and (6), we obtain M.sub.M values of 46.09 g/mol and 43.738 g/mol, respectively, i.e., a discrepancy of 5.38%. Significantly, the discrepancy between these equations increases for mixtures of gases having higher molecular weight ratios.
Existing mixing rules for calculating .gamma..sub.M values also lead to substantially different results. See Hallewell et al. "A simplified formula for the analysis of binary gas containing a low concentration of a heavy vapor in a lighter carrier", Ultrasonics Symposium, Proceedings, New York, Institute of Electrical and Electronic Engineers, 1994, 1311-1316.
For purposes of comparison with the mixing rule of the present invention, the following formulation of equation (4) will be used since it represents ion most commonly used by prior art workers: ##EQU8##
In addition to the discrepancies resulting from different mixing rules, it is also important to note that all prior art methods of acoustical concentration measurement apply to real gases equations which were developed for ideal gases, i.e., they use the ideal gas constant R which implicitly assumes that the compressibility factor of the gas is equal to one. This assumption can cause an error of up to 5% (sometimes even higher) in the speed of sound values obtained from equations (1), (4), or (7). It can also cause errors in the evaluation of the specific heat capacities used in these equations.
For pure gases, this problem can potentially be solved by substitution of R*=zR for R in equations (1) and (2), where z is a compressibility factor. As discussed in Reid et al., supra, at small deviations from ideal gas behavior z can be obtained from the formula: ##EQU9## where P.sub.r =P/P.sub.c is relative pressure, T.sub.r =T/T.sub.c is relative temperature, P is actual pressure, T is actual temperature, .omega. is an "acentric" factor, T.sub.c is the gas' critical temperature, and P.sub.c is its critical pressure, where .omega., T.sub.c, and P.sub.c are constants for the pure gas. For gases near saturation conditions, Redlich-Kwong, Soave or Peng-Robinson approaches can be used. (See Reid et al., supra, at pages 42-46.)
This problem of compressibility, of course, is much more complicated for a gas mixture since rules for determining the compressibility factor of a gas mixture have not been reported in the literature.
(2) Lack of Known Thermodynamic Properties for Many gases.
From equations (1)-(4), we can see that estimation of concentrations requires a priori knowledge of specific heat capacity at both constant pressure and constant volume as a function of temperature.
Coefficients A.sub.c, B.sub.c, C.sub.c, and D.sub.c of equation (3) can be used to calculate c.sub.p, and these coefficients have been published for some chemicals. Very often, however, the coefficients have not been published and, indeed, even the manufacturer of the chemical of interest may not have the information needed to estimate the coefficients. For these cases, under the state of the art prior to present invention, it has been impossible to use acoustical methods to measure concentrations.
(3) Insufficient Accuracy of Existing Methods and Apparatus for Measuring The Speed of sound in a gas mixture.
Known acoustic methods and instruments typically employ the shadow acoustic method in which separate transducers are used for transmitting and receiving ultrasonic waves. This approach permits the use of either flexure or radial mode transducers, which have some advantages in comparison to thickness mode transducers, especially for gaseous medium applications. In particular, such transducers have significantly higher electroacoustic efficiency and smaller dimensions (see Folkestad et al. "Chirp Excitation of Ultrasonic Probes and Algorithm for Filtering Transit Times in High-Rangeability Gas Flow Metering", IEEE Transactions on Ultrasonics, Ferroelectrics and Frequency Control, Vol. 40, No. 3, May 1993, 193-215).
The use of separate transducers, however, does have a significant disadvantage which has not been effectively addressed in the prior art. That disadvantage arises from the fact that the acoustic and electronic paths of the transmitted and the received signals are different and thus significant errors can be introduced into the speed of sound determination as a result of:
(a) Spectrum distortion and time delay of the first half-wave of the received signal in the piezoelements, in the transfer layers between the piezoelements and the gaseous medium, and in the electronic networks of the transmitter and receiver. Since these distortions and delays are constant for any given hardware system, this type of systematic error can at least in theory be compensated for during calibration using an ideal or near ideal gas, provided the distance between the two transducers is known to within a few micrometers.
(b) Spectrum distortion of the first half-wave of the received signal as result of operating under far-field conditions as is normally done for systems using the shadow method. That is, the transmitter and the receiver are spaced from one another by an acoustical path length L.sub.p which satisfies the relationship: ##EQU10## where d is a diameter of the transducer and .lambda. is the wavelength of the acoustical energy given by: ##EQU11## where C is the speed of sound in the testing media and .function. is the transducer's operating frequency.
Operating in the far-field zone means that an orientation diagram (acoustic field plot) for the transmitter will generally have a spherical shape. More importantly with regard to concentration measurements, operating in the far-field zone means that the orientation diagram will be different for gases with different acoustical properties. As a result, different gas mixtures will produce different distortions of the first half-wave of the received signal, as well as different variations in its duration. Plainly, this type of systematic error is dependent upon the acoustical properties of the gases in the mixture as well as their concentrations, and thus cannot be compensated for during calibration.
(c) Spectrum distortion of the first half-wave of the acoustic signal by the medium which is being tested. The absorption coefficient .alpha. of an ideal gas can be estimated from the combined Stokes-Kirchhoff's expression (see Hunter "Acoustics", Englewood Cliffs, N.J., Prentice-Hall, Inc., 1957) ##EQU12## where .gamma. is the gas' specific heat ratio, .eta. is its viscosity, k is its thermal conductivity, P is pressure, C is the speed of sound, and C.sub.p is thermal capacitance defined as c.sub.p /M where c.sub.p is the heat capacity at constant pressure and M is molecular weight.
From this equation, it can be seen that gases function as high order low-pass filters, with the characteristics of the filter being dependent on temperature and pressure. The behavior of the filter becomes even more complex for real polyatomic gases, where molecular relaxation effects come into play (see Bhatia, "Ultrasonic Absorption", Dover Publications, Inc., New York, 1985).
As with the far-field effect, the filtering effect depends on the composition and concentrations of the gas mixture, with different mixtures producing different distortions of the first half-wave of the received signal and different changes in its duration. Once again, this type of systematic error cannot be compensated for during calibration.
(d) Spectrum distortion of the first half-wave of the received signal due to side wall reflection interference. This effect occurs during testing of a gas mixture with a low speed of sound. Acoustic chambers are clearly not semi-infinite and thus there are opportunities for the reflection of an acoustic wave having a spherical orientation diagram from the side walls, followed by interference of the reflected wave with the original wave front at the surface of the receiver. This type of systematic error depends upon the acoustical properties of the gas mixture, i.e., the gases making up the mixture and their concentrations.
Of the foregoing four types of systematic error, only the first type is independent of the properties of the gas being tested and can be compensated for during calibration with an ideal or near ideal gas. The other three types of systematic error, which together can achieve several percent, are dependent on the acoustical properties of the gas being tested, and can change significantly with temperature variations. Their reduction is possible only by frequent re-calibration of the instrument with different gases at different concentrations and temperatures, which is clearly undesirable.
D. Summary of the State of the Art
From the foregoing, it can be seen that the existing techniques for performing acoustical measurements suffer from numerous drawbacks and limitations. In particular, all the existing acoustical concentration measurement methods and instruments require frequent maintenance to reduce the total error introduced by different recipes, as well as by temperature and pressure variations. In addition, these techniques are associated with significant inaccuracies and ambiguities in the concentration measurement, which limit their usefulness in producing highly accurate and reliable measurements.